A067701 -id:A067701 - cp's OEIS frontend

A066232 Numbers k such that phi(k) = phi(k-2) - phi(k-1).

195, 3531, 9339, 27231, 46795, 78183, 90195, 112995, 135015, 437185, 849405, 935221, 1078581, 1283601, 1986975, 2209585, 2341185, 2411175, 2689695, 2744145, 3619071, 3712545, 4738185, 5132985, 6596121, 7829031, 8184715, 12176109

Offset: 1

Joseph L. Pe, Dec 18 2001

nonn

As in A065557, all terms listed here are odd. Problem: Prove that this holds in general.

			Phi(195) = 96 = 192-96 = phi(193)-phi(194).

Harry J. Smith and Jud McCranie, Table of n, a(n) for n = 1..289 (first 55 terms from Harry J. Smith)

Cf. A065557, A066231, A067701, A197112, A220160.

Select[Range[3, 10^6], EulerPhi[ # ] == EulerPhi[ # - 2] - EulerPhi[ # - 1] &]

isok(k) = { k > 2 && eulerphi(k) == eulerphi(k - 2) - eulerphi(k - 1) } \\ Harry J. Smith, Feb 07 2010

a(n) = A220160(n) + 1 = A197112(n) + 2. - Andrew Howroyd, Dec 19 2024

a(13)-a(28) from Harry J. Smith, Feb 07 2010

A066231 Numbers n such that phi(n) = phi(n-1) - phi(n-2).

Original entry on oeis.org

6, 8, 26, 78, 218, 306, 3666, 4646, 5066, 8816, 12206, 12546, 19878, 20436, 24236, 29546, 37736, 47996, 60116, 72086, 73026, 77046, 87476, 121146, 126056, 129246, 149756, 190268, 234636, 247856, 273296, 275724, 419366, 531236, 553476, 621726

Offset: 1

Joseph L. Pe, Dec 18 2001

nonn

Question: Are all terms of this sequence even? (Compare A065557, whose terms could be all odd and squarefree.)

			phi(8) = 4 = 6-2 = phi(7) - phi(6).

Harry J. Smith and Jud McCranie, Table of n, a(n) for n = 1..494 (first 117 terms from Harry J. Smith)

Cf. A066232, A067701.

Flatten[Position[Partition[EulerPhi[Range[630000]],3,1],?(#[[3]] == #[[2]]- #[[1]]&),1,Heads->False]]+2 (* _Harvey P. Dale, Aug 19 2018 *)

n=0; for (m=3, 10^9, if (eulerphi(m) == eulerphi(m - 1) - eulerphi(m - 2), write("b066231.txt", n++, " ", m); if (n==117, return)) ) \\ Harry J. Smith, Feb 06 2010

a(24)-a(36) from Harry J. Smith, Feb 06 2010

A067843 Least solution k>n of phi(k-n)+phi(k+n)=phi(2k).

Original entry on oeis.org

5, 10, 7, 10, 11, 12, 35, 14, 13, 22, 55, 22, 19, 70, 19, 22, 85, 26, 77, 26, 27, 110, 55, 34, 55, 38, 31, 34, 119, 38, 65, 44, 41, 52, 65, 46, 185, 154, 43, 46, 143, 54, 215, 70, 57, 110, 161, 58, 187, 68, 67, 76, 203, 62, 175, 62, 61, 76, 95, 74, 67, 130, 71, 88, 95, 82, 115, 74, 73, 130, 215

Offset: 1

Joseph L. Pe, Feb 11 2002

nonn

From Robert Israel, Jun 08 2018: (Start)
The first n for which a(n)-n is odd is 239.
If n+2 and n+4 are twin primes (i.e. n+2 is in A001359), then a(n) <= n+4.
Conjecture: a(n) >= n+4 for all n. (End)

			k = 10 is the smallest solution of phi(k-2)+phi(k+2)=phi(2k). So a(2) = 10.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A067701.

f:= proc(n) local k;
     for k from n+1 do if numtheory:-phi(k-n)+numtheory:-phi(k+n)=numtheory:-phi(2*k) then return k fi od:
end proc:
map(f, [$1..100]); # Robert Israel, Jun 08 2018

f[k_] := Module[{i = k + 1}, While[EulerPhi[i - k] + EulerPhi[i + k] != EulerPhi[2 i], i++ ]; i]; Table[f[n], {n, 1, 40}]

More terms from Robert Israel, Jun 08 2018

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A066232 Numbers k such that phi(k) = phi(k-2) - phi(k-1).

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A066231 Numbers n such that phi(n) = phi(n-1) - phi(n-2).

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A067843 Least solution k>n of phi(k-n)+phi(k+n)=phi(2k).

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